Concerted Efforts? Monetary Policy and Macro-Prudential Tools

Transcription

1 Concerted Efforts? Monetary Policy and Macro-Prudential Tools Andrea Ferrero Richard Harrison Benjamin Nelson University of Oxford Bank of England Centre for Macroeconomics April 6, 218 Abstract The inception of macro-prudential policy frameworks in the wake of the global financial crisis raises questions of how macro-prudential and monetary policies should be coordinated. We examine these questions through the lens of a macroeconomic model featuring nominal rigidities, housing, incomplete risk-sharing between borrower and saver households, and macro-prudential tools in the form of mortgage loan-to-value and bank capital requirements. We derive a welfare-based loss function which suggests a role for active macro-prudential policy to enhance risk-sharing. Macro-prudential policy faces tradeoffs, however, and complete macro-prudential stabilization is not generally possible in our model. Nonetheless, simulations of a house price boom and subsequent correction suggest that macro-prudential tools could alleviate debt-deleveraging and help avoid zero lower bound episodes, even when macro-prudential tools themselves impose only occasionally binding constraints on debt dynamics in the economy. Preliminary and incomplete, please do not circulate without permission. The views expressed are those of the authors and should not be taken to represent the views of the Bank of England or any of its policy committees. We thank participants at the second annual ECB macro-prudential policy and research conference, the CEPR MMCN 217 conference, the Economic Growth and Policy Conference 217, the 5th Macro Banking and Finance Workshop, the 2th Central Bank Macroeconomic Modeling Workshop on Policy Coordination, our discussants Larry Christiano and Christian Friedrich and seminar participants at Oxford University for helpful comments. We are grateful to Andy Blake, Stefan Laséen, Lien Laureys, Roland Meeks and Matt Waldron for useful discussions on previous versions of this paper. Ferrero: andrea.ferrero@economics.ox.ac.uk; Harrison: richard.harrison@bankofengland.co.uk; Nelson: benjamindnelson@hotmail.com 1

2 1 Introduction With the recovery in the UK economy broadening and gaining momentum in recent months, the Bank of England is now focussed on turning that recovery into a durable expansion. To do so, our policy tools must be used in concert. Carney (214) In the aftermath of the global financial crisis, macro-prudential policy frameworks have been established and developed across the world. The inception of these frameworks raises questions of how new policy tools should be operated and how macro-prudential and monetary policies should be coordinated. These questions have particular resonance given the conditions currently facing policymakers in many advanced economies. Since the financial crisis, monetary policy has been set with a view to supporting economic activity and hence preventing inflation from falling below target. That has required a prolonged period of low, sometimes negative, short-term policy rates and a raft of so-called unconventional monetary policy measures. These policies have supported asset prices and kept borrowing costs low. However, these effects have also given rise to concerns that such monetary conditions may lead to levels of indebtedness that threaten financial stability. In some cases, macro-prudential policy instruments have been used to guard against these risks. So the policy mix in many economies could be crudely summarized as loose monetary policy and tight macro-prudential policy. 1 In this paper, we examine these questions through the lens of a simple and commonly used modeling framework. Our model is rich enough to generate meaningful policy tradeoffs, but sufficiently simple to deliver tractable expressions for welfare and analytical results in some cases. Our model incorporates borrowing constraints and nominal rigidities. These frictions give rise to meaningful roles for macro-prudential policy and monetary policy respectively. financial friction takes the form of a collateral constraint, following Kiyotaki and Moore (1997) and Iacoviello (25) among many others. The The collateral constraint limits the amount that relatively impatient households can borrow. Specifically, their debt cannot exceed a particular fraction of the value of the housing stock that they own: there is a loan to value (LTV) constraint which the macro-prudential authority can vary. In turn, borrowing by relatively impatient households is financed by saving by more patient households ( savers ). A perfectly competitive banking system intermediates the flow of saving from savers to borrowers. Banks are subject to a capital requirement and we assume that raising equity is costly (see also Justiniano et al., 214). As such, variation in capital requirements generates fluctuations in the spread between borrowing and deposit rates, providing the authorities with a second macro-prudential tool in addition to the LTV, albeit one that is costly to deploy. Finally, the nominal frictions 1 The quote at the start of the paper is from the opening statement at a press conference explaining the decision of Bank of England s Financial Policy Committee to limit the quantity of new lending at high loan-to-value ratios. That statement explains that: The existence of macro-prudential tools allows monetary policy to focus on its primary responsibility of price stability. In other words, monetary policy does not need to be diverted to address a sector-specific risk in the housing market. Similarly, authorities in Canada have tightened macro-prudential policy several times since the financial crisis (see Krznar and Morsink, 214) while the official policy rate has remained low. Conversely, the policy mix in Sweden has been a subject of much controversy and debate (see, for example, Jansson, 214; Svensson, 211). 2

3 are Calvo (1983) contracts, standard in the New Keynesian literature. We derive a welfare-based loss function as a quadratic approximation to a weighted average of the utilities of borrowers and savers. The loss function has five (quadratic) terms. Two terms stem from the nominal rigidities in the model and are familiar from New Keynesian models: the policymaker seeks to stabilize the output gap and inflation. The remaining terms are generated by the financial friction: the policymaker seeks to stabilize the distribution of non-durable consumption and housing consumption between borrowers and savers the consumption gap and the housing gap respectively. The presence of household heterogeneity therefore gives rise to objectives whose origin lies in the incompleteness of risk sharing between households in the economy. The final term in the loss function captures the costs of varying capital requirements, which themselves stem from the non-zero cost of equity we assume outside of steady state. We use the model to study how monetary and macro-prudential policies should optimally respond to shocks. To build some intuition, we first focus on a linear approximation of the model around a steady state in which the borrowing constraint is always binding and the full value of housing can be used as collateral. We demonstrate that macro-prudential policy generally faces a tradeoff in stabilizing the distribution of consumption and the distribution of housing services even when prices are flexible and both macro-prudential tools are used. We also show that monetary policy alone has relatively little control over these distributions, particularly the distribution of housing between borrowers and savers. In other words, imperfect risk sharing is a real phenomenon whose consequences could be addressed by macro-prudential policies, but these policies also imply costs that must be accounted for when deploying them. This tradeoff prevents complete macro-prudential stabilization using the tools we study, even under flexible prices. Index (Jan 2 = 1) Real house price Real secured debt (a) Level % deviation from trend Real house price Real secured debt (b) Linear de trended Figure 1: UK house prices and mortgage debt To examine the quantitative implications of the model for optimal monetary and macroprudential policy, we explore numerical experiments designed to simulate a housing boom and subsequent house price correction, calibrated with reference to the experiences of the United 3

4 Kingdom and the United States in the decades preceding and following 28 (Figure 1). This allows us to examine the extent to which macro-prudential tools could have complemented monetary policy in achieving macroeconomic stabilization goals in the face of a house price fall large enough to force the nominal rate to the zero bound. We find that macro-prudential policies, in the form of the LTV tool and bank capital requirements, allow for better stabilization of the consumption and housing gaps, but also allow monetary policy to fully stabilize the output gap and inflation because the short-term nominal interest rate does not hit the zero bound. In other words, during a house price correction the optimal conduct of macro-prudential policies mitigates the fall in the equilibrium real rate of interest (that is, the real interest rate consistent with closing the output gap Ferguson, 24). When the LTV tool is not available to the policymaker, the behavior of policy during the recovery from a housing bust is broadly consistent with the loose monetary policy and tight macro-prudential policy mix observed in many economies in recent years. In that case, following an initial cut in response to the fall in house prices, capital requirements are progressively tightened. This macro-prudential policy behavior slows the speed of de-leveraging and ensures that the equilibrium real interest rate recovers more quickly than otherwise. The faster recovery of the equilibrium real interest rate allows monetary policy to liftoff from the zero bound earlier than would otherwise have been the case. However, after liftoff the optimal monetary policy stance is slightly stimulative, to cushion the aggregate demand effects from tightening capital requirements. Our results contribute to a growing literature exploring the conduct and coordination of macro-prudential policy. In the context of optimal policy, Angelini et al. (212), Bean et al. (21) and de Paoli and Paustian (217) consider the coordination between monetary and macro-prudential policies in models with similar frictions to ours. Those papers also find that there are cases in which it is optimal to adjust monetary and macro-prudential policy instruments in opposite directions. The welfare-based loss function in our model is similar to that derived by Andres et al. (213) in a similar model, and bears similarity to Curdia and Woodford (21). However, those authors focus on the analysis of optimal monetary policy and do not explore macro-prudential policy. Other papers with a greater focus on macro-prudential policies include Angeloni and Faia (213), Gertler et al. (212), Clerc et al. (215), Christiano and Ikeda (216) and Aikman et al. (218). The focus of each of these papers is on macro-prudential bank capital instruments, whereas we also consider a macro-prudential LTV tool. The potential of using LTV instruments is of particular relevance for countries like the UK, where mortgage lending is the single largest asset class on domestic banks balance sheets, and is also the single largest liability class on households balance sheets. Finally, Eggertsson and Krugman (212) study the implications of households debt deleveraging for monetary policy in a model similar to ours, while Korinek and Simsek (216) and Farhi and Werning (216) study the theoretical implications of the zero bound constraint on monetary policy and distortions in financial markets for optimal macro-prudential policies. The rest of the paper is organised as follows. Section 2 presents the model. Section 3 derives a linear-quadratic approximation of the equilibrium and discusses some analytical results. Section 4

5 4 illustrates the optimal joint conduct of monetary and macro-prudential policy via numerical simulations. Section 5 concludes. 2 Model The economic agents in the model are households, banks, firms, and the government. Households are heterogeneous in their degree of patience. Banks transfer funds from savers to borrowers and fund their operations with a mix of deposits and equity. Firms produce goods for consumption. The government conducts monetary and macro-prudential policy. 2.1 Households Patient households (i.e. savers, indexed by s) have a higher discount factor than impatient households (i.e. borrowers, indexed by b). We denote with ξ (, 1) the mass of borrowers, and normalise the total size of the population to one. We also assume perfect risk sharing within each group Savers A generic saver household i, 1 ξ) decides how much to consume in goods C s t (i) and housing services H s t (i), 2 save in deposits D s t (i) and equity E s t (i) of financial intermediaries, and work L s t(i), to maximise W s (i) E { t= β t s ]} ( 1 e zcs(i)) t + χs H e uh t (Ht s (i)) 1 σ h χs L 1 σ h 1 + ϕ (Ls t(i)) 1+ϕ, (1) where β s (, 1) is the individual discount factor, z > is the degree of absolute risk aversion, σ h is the inverse elasticity of housing demand, ϕ is the inverse Frisch elasticity of labour supply, and χ s H, χs L > are type-specific normalisation constants. Preferences include an aggregate housing preference shock, u h t, common to all households. The budget constraint for patient household i is P t C s t (i) + D s t (i) + E s t (i) + (1 + τ h )Q t H s t (i) = W s t L s t(i) + R d t 1D s t 1(i) + R e t 1E s t 1(i) + Q t H s t 1(i) + Ω s t(i) T s t (i) Γ t (i), where P t is the consumption price index, Q t is the nominal house price, W s t is the nominal wage for savers, Rt 1 d is the nominal return on bank deposits, and Re t 1 is the nominal return on bank equity. 3 The variable Tt s (i) captures lump-sum taxes while Ω s t(i) denotes the savers share of remunerated profits from intermediate goods producers and from banks. The constant τ h is a tax/subsidy on savers housing that we assume is set to deliver an efficient steady state in the housing market. The final term in the budget constraint is a cost associated with deviations 2 We make the standard assumption that the flow of housing services is proportional to the stock of housing. 3 As in Benigno et al. (214), the introduction of type-specific wages and exponential utility simplifies aggregation, and facilitates the derivation of a welfare criterion for the economy as a whole. 5

6 from some preferred portfolio level of bank equity κ > Γ t (i) Ψ 2 Et s (i) κξdt b 1 /(1 ξ) ]2 κξd b t 1 ξ, with Ψ >. For analytical convenience, we express the adjustment cost relative to aggregate bank lending ξd b t, which savers take as given Borrowers A generic borrower household i 1 ξ, 1] maximizes the same per-period utility as savers (1) but discounts the future at lower rate β b (, β s ). The borrower s budget constraint is P t C b t (i) D b t(i) + Q t H b t (i) = W b t L b t(i) R b t 1D b t 1(i) + Q t H b t 1(i) + Ω b t(i) T b t (i), where D b t(i) is the amount of borrowing at time t, T b t (i) are lump-sum taxes, including those used to obtain an efficient allocation of consumption in the model s steady state, and Ω b t(i) denotes profits from ownership of intermediate goods producing firms. As common in the literature (e.g Kiyotaki and Moore, 1997), we assume that a collateral constraint limits impatient households ability to borrow. In particular, their total liabilities cannot exceed a (potentially time-varying) fraction of their current housing wealth D b t(i) Θ t Q t H b t (i), where Θ t, 1]. The term Θ t represents the maximum loan-to-value (LTV) ratio available to borrowers. The standard interpretation of such a constraint is that lenders (in this case, the financial intermediaries) require borrowers to have a stake in a leveraged investment to prevent moral hazard behavior. In our policy analysis, we will consider cases in which the macro-prudential authority sets the maximum LTV that banks can extend to borrowers. In this sense, the LTV ratio is part of the macro-prudential toolkit that we will study below. 2.2 Banks A continuum of perfectly competitive banks, indexed by k, 1], raise funds from savers in the form of deposits and equity (their liabilities), and make loans (their assets) to borrowers. Bank k s balance sheet identity is D b t(k) = D s t (k) + E s t (k). (2) In addition, we assume that equity must account for at least a fraction κ t of the total amount of loans banks extend to borrowers E s t (k) κ t D b t(k). (3) 4 The introduction of this adjustment cost function is a simple way to distinguish bank equity from bank debt (deposits), and captures the idea that deposits are generally more liquid, and thus easier for households to adjust. Little of substance would change in the first-order accurate solution to the model that we examine if we specified bank equity as a state-contingent claim. 6

7 The presence of equity adjustment costs breaks down the irrelevance of the capital structure (the Modigliani-Miller theorem). Savers demand a premium for holding equity, which banks pass on to borrowers in the form of a higher interest rate. From the perspective of the bank, equity is expensive, and thus deposits are the preferential source of funding. In the absence of any constraint, banks would choose to operate with zero equity and leverage would be unbounded. Equation (3) ensures finite leverage for financial intermediaries. As in the case of the LTV parameter, we will consider two possible interpretations of κ t. The first treats this variable as exogenous, relying on the notion that financial institutions target a certain leverage ratio due to market forces (Adrian and Shin, 21). According to the second interpretation, while the constraint still plays the role of limiting banks leverage, it is the macro-prudential authority that sets the capital requirement on financial institutions. In this sense, κ t becomes the second macro-prudential tool for the regulatory authority. Several recent contributions have discussed capital requirements as one of the key instruments to avoid financial crises in the future (e.g. Admati and Hellwig, 214; Miles et al., 213). In our analysis, we will focus on the interaction between capital requirements and interest rate setting. Independently of its interpretation, the capital requirement constraint is always binding in equilibrium, exactly because financial intermediaries seek to minimize their equity requirement. If the capital constraint of all banks were slack, one bank could marginally increase its leverage, charge a lower loan rate, and take the whole market. Therefore, competition drives the banking sector against the constraint. Banks profits are P t (k) R b td b t(k) R d t D s t (k) R e t E s t (k) = R b t (1 κ t )R d t κ t R e t ]D b t(k), where the second equality follows from substituting the balance sheet constraint (2) and the capital requirement (3) at equality. The zero-profit condition implies that the loan rate is a linear combination of the return on equity and the return on deposits R b t = κ t R e t + (1 κ t )R d t, where the time-varying capital requirement represents the weight on the return on equity. A surprise rise in κ t, whether due to an exogenous shock or a policy decision, forces banks to delever and raises credit spreads. 2.3 Production A representative retailer combines intermediate goods according to a technology with constant elasticity of substitution ε > 1 1 Y t = ] ε Y t (f) ε 1 ε 1 ε df, 7

8 where Y t (f) represents the intermediate good produced by firm f, 1]. Expenditure minimisation implies that the demand for a generic intermediate good is ] Pt (f) ε Y t (f) = Y t, (4) where P t (f) is the price of the variety produced by firm f and the aggregate price index is 1 P t = P t ] 1 P t (f) 1 ε 1 ε df. Intermediate goods producers operate in monopolistic competition, are owned by savers and borrowers according to their shares in the population, and employ labour to produce variety f according to Y t (f) = A t L t (f). (5) Aggregate technology A t follows a stationary autoregressive process in logs ln A t a t = ρ a a t 1 + ɛ a t, with ρ a (, 1) and ɛ a t N (, σ 2 a). To simplify aggregation, we assume L t (f) is a geometric average of borrower and saver labour, with weights reflecting the shares of the two types and the corresponding wage index is L t (f) L b t(f)] ξ L s t(f)] 1 ξ, W t (W b t ) ξ (W s t ) 1 ξ. Intermediate goods producers set prices on a staggered basis. As customary, we solve their optimization problem in two steps. First, for given pricing decisions, firms minimize their costs, which implies that the nominal marginal cost M t is independent of each firm s characteristics. The second step of the intermediate goods producers problem is to determine their pricing decision. As in Calvo (1983), we assume firms reset their price P t (f) in each period with a constant probability 1 λ, taking as given the demand for their variety, while the complementary measure of firms λ keeps their price unchanged. The optimal price setting decision for firms that do adjust at time t solves max P t(f) { } E t λ i Q t,t+v (1 + τ p ) P t (f) M t+v ]Y t+v (f), v= subject to (4), where τ p is a subsidy to make steady state production efficient. Intermediate goods producers are owned by households of both types in proportion to their shares in the population. Therefore, we assume that the discount rate for future profits is Q t,t+v (Q b t,t+v) ξ (Q s t,t+v) 1 ξ, 8

9 where Q j t,t+v = β jze z(cj t+v (i) Cj t (i)) is the stochastic discount factor between period t and t + v of type j = {b, s}. 2.4 Equilibrium Because of the assumption of risk-sharing within each group, all households of a given type consume the same amount of goods and housing services and work the same number of hours. Therefore, in what follows, we drop the index i and characterize the equilibrium in terms of type-aggregates. Similarly, because all financial intermediaries make identical decisions in terms of interest rate setting, we drop also the index k and simply refer to the aggregate balance sheet of the banking sector. For a given specification of monetary and macro-prudential policy, an imperfectly competitive equilibrium for this economy is a sequence of quantities and prices such that households and intermediate goods producers optimise subject to the relevant constraints, final good producers and banks make zero profits, and all markets clear. 5 In particular, for the goods market, total production must equal the sum of consumption of the two types plus the resources spent for portfolio adjustment costs 6 where Γ t Y t = ξc b t + (1 ξ)c s t + Γ t, (6) 1 ξ Γ t (i)di = Ψ 2 ( ) 2 κt κ 1 κξdt. b We assume housing is in fixed supply (i.e., land). The housing market equilibrium then requires H = ξh b t + (1 ξ)h s t, (7) where H is the total available stock of housing. 7 Finally, in the credit market, total bank loans must equal total household borrowing. Thus, the aggregate balance sheet for the financial sector respects ξd b t = (1 ξ)(d s t + E s t ), where per-capita real private debt (derived from the borrowers budget constraint) evolves according to D b t P t = Rb t 1 Π t D b t 1 P t 1 + C b t Y t + Q t P t (H b t H b t 1) + T b, and T b is a subsidy that ensures the steady state allocation is efficient. 3 Linear-Quadratic Framework Our ultimate objective is to study the optimal joint conduct of monetary and macro-prudential policy. In this section, we aim to obtain some analytical results following the approach of the 5 Appendix A reports the equilibrium conditions for the private sector and the details of aggregation. 6 The resource constraint follows from combining the budget constraints of the two types (aggregated over their respective measures) with the financial intermediaries balance sheet, under the assumption that the government adjusts residually the lump-sum transfers to savers. 7 The absolute level of this variable plays no role in the analysis. 9

10 optimal monetary policy literature (e.g. Clarida et al., 1999; Woodford, 23), and derive a linear-quadratic approximation to our model with nominal rigidities and financial frictions. We approximate the model around a zero-inflation steady state in which the collateral constraint binds. An appropriate choice of taxes and subsidies ensures that the steady state allocation is efficient. Appendix C reports the details of the derivations. 3.1 Quadratic Loss Function To derive the welfare-based loss function, we take the average of the per-period utility functions of borrowers and savers, weighting each type according to their share in the population. We assume that policymakers discount the future at rate β s. 8 A second-order approximation of the resulting objective gives L 1 2 E t= β t s (x 2 t + λ π π 2 t + λ κ κ 2 t + λ c c 2 t + λ h h2 t ), (8) where lower-case variables denote log-deviations from the efficient steady state, x t y t y t is the efficient output gap (with yt representing the efficient level of output), c t c b t c s t is the consumption gap between borrowers and savers, and h t h b t h s t is the housing gap between borrowers and savers. target are The weights on deviations of inflation and capital requirements from λ π ε γ and λ κ ψη σ + ϕ, where γ (1 βλ)(1 λ)(σ + ϕ)/λ, while the weights on the consumption and housing gaps are λ c ξ(1 ξ)σ(1 + σ + ϕ) (1 + ϕ)(σ + ϕ) and λ h σ hξ(1 ξ) σ + ϕ. The loss function (8) features two sets of terms. The first includes the efficient output gap and inflation the standard variables that appear in the welfare-based loss function of a large class of New Keynesian models. Their presence in the loss function reflects the two distortions associated with price rigidities. First, such rigidities open up a labour wedge, causing the level of output to deviate from its efficient level. Second, staggered price setting implies an inefficient dispersion in prices, which is proportional to the rate of inflation. The second set of terms in (8), comprising the consumption gap and the housing gap, arise from the heterogeneity between household types and, in particular, from the fact that one group of households are credit-constrained while the others are not. The collateral constraint generates an inefficiency because of incomplete insurance. In the absence of the collateral constraint, households could insure each other against variation in their housing and consumption bundles. The collateral constraint limits the amount of borrowing that can take place to carry out this 8 Benigno et al. (214) assume that the discount factor of the two types is the same in the limit (β b β s), and that borrowing/lending position are determined by the initial distribution of wealth. We retain the heterogeneity in the discount factors but effectively assume that the policymaker is chosen among the population of savers. This choice is obviously arbitrary but we can solve the optimal policy problem for any value of β (, 1). 1

11 insurance in full. As a result, risk sharing is imperfect. An analogous argument applies to housing. Imperfect risk sharing therefore becomes a source of welfare losses the policymaker accounts for when setting policy optimally. Finally, the term λ κ κ 2 t accounts for the costs associated with the use of capital requirements as a policy tool Linearised Constraints In this section, we combine the linearised equilibrium relations to obtain a parsimonious set of constraints for the optimal policy problem in our linear-quadratic setting. 1 To simplify the derivations, we assume Θ = 1 (a 1% LTV ratio). We return to the case Θ < 1 in the quantitative analysis. Appendix D provides additional details on the derivations. On the supply side, as common in the literature, we can rewrite the Phillips curve in terms of the efficient output gap π t = γx t + βe t π t+1 + u π t, (9) where u π t is an exogenous cost-push shock. On the demand side, we write an aggregate demand curve in terms of the output gap and the consumption gap x t ξ c t = σ 1 (i t E t π t+1 ) + E t (x t+1 ξ c t+1 ) + ν c t, (1) where ν c t is a combination of exogenous shocks defined in the appendix. In a standard representative agent model, the consumption gap is zero, and all agents behave like the savers in our economy. The consumption gap in (1) summarises the impact of debt obligations, house prices and LTV ratios on aggregate demand due to the lack of risk sharing. We derive an equation for the housing gap by taking the difference of the housing demand equations between borrowers and savers. The resulting expression is h t = ω ξ(β s β b ) (i t E t π t+1 ) + β s β b σ h ξω σ h ω (q t E t q t+1 ) σ σ h ξ (x t E t x t+1 ) + σ σ h c t + µ σ h ω θ t 1 µ σ h ω ψκ t + ν h t, (11) where ω and νt h are combinations of fundamental parameters and shocks, respectively, defined in the appendix. To complete the description of the housing block, we take a population-weighted average of the housing demand equation to obtain an aggregate house price equation that reads as q t = (i t E t π t+1 ) + σω ω + β E tx t+1 + ξ µ ω + β θ ξ(1 µ) t ω + β ψκ t + β ω + β E tq t+1 + ν q t, (12) where ν q t is a combination of fundamental shocks defined in the appendix. 9 If we treat the leverage ratio as an exogenous albeit time-varying constraint, the costs of its fluctuations become independent of policy, and thus irrelevant for ranking alternative policies in terms of welfare. In this case, the welfare objective only has four terms but the policymaker has one fewer tool for stabilization purposes. 1 Unless otherwise stated, lower-case variables denote log-deviations from steady state. For a generic variable Z t, with steady state value Z, z t ln(z t/z). 11

12 In the neighborhood of a steady state in which the borrowing constraint binds, debt is a function of the LTV constraint, house prices and the housing gap d b t = θ t + q t + (1 ξ) h t. (13) We can keep track of its dynamics via the borrowers budget constraint d b t = 1 (i t 1 + ψκ t 1 + d b t 1 π t ) + (1 ξ)( h t β h t 1 ) + 1 ξ c t. (14) s η The set of endogenous states in the model consists of private debt, the nominal interest rate, and the leverage ratio. Since we solve the optimal policy problem under discretion, with endogenous state variables, we need to keep track of the effects of current actions on future losses through the evolution of the states. In this respect, we can simplify the optimal policy problem by defining a single composite state variable S t d b t + i t + ψκ t β s (1 ξ) h t, (15) which captures the burden of debt at maturity for borrowers relative to the discounted quantity of housing owned. Using this composite state variable, we can rewrite the borrowing constraint at equality as S t = θ t + q t + i t + ψκ t + (1 β s )(1 ξ) h t. (16) Similarly, the law of motion of debt can be rewritten as the law of motion of the single state variable as S t = 1 (S t 1 π t ) + i t + ψκ t + (1 β s )(1 ξ) β h t + 1 ξ c t. (17) s η In this way, we have reduced the endogenous states from three to one, and we can characterise the effects of current decisions on future losses via the variable S t only. The joint optimal monetary and macro-prudential problem under discretion consists of minimizing (8) subject to the constraints (9)-(14), or, equivalently, (9)-(12), (16), and (17). general, the policymaker can choose three instruments (the nominal interest rate, i t, the LTV ratio, θ t, and the capital requirement, κ t ) to implement the optimal plan. To fix ideas before deriving optimal policy plan in the most general case, however, we start from a simple benchmark without nominal rigidities in which debt is issued in real terms. This special case allows us to focus on the characteristics of macro-prudential policy, abstracting from its interactions with monetary policy. 3.3 Optimal Macro-Prudential Policy under Flexible Prices To highlight the effects of macro-prudential policy, we focus on the efficient equilibrium of the model. With flexible prices (λ ) and no markup shocks (u m t =, t), productivity fully determines output, which becomes exogenous (y t = yt ), so that the output gap is always zero (x t =, t). In addition, the weight on inflation in (8) is zero (λ π = ). Hence, the first two terms in the loss function disappear. The job of the policy authority then becomes to stabilize the consumption and housing gap, and if used as an instrument minimize the volatility of In 12

13 capital requirements. The Phillips curve is no longer a constraint for the optimal policy plan. Since the output gap is always zero, the aggregate demand curve determines the consumption gap as a function of the real interest rate r t ξ c t = σ 1 r t + ξe t c t+1 ν c t. The other constraints correspond to equation (11), (12), (16), and (17), after imposing a zero output gap in all periods. In addition, because of the assumption that debt is in real terms, inflation disappears from the model and the nominal interest rate is replaced by the real interest rate. Appendix E.1 sets up the Lagrangian that describes the optimal policy problem under discretion and derives the first order conditions. After solving out for the Lagrange multipliers, we can express the optimal policy plan in terms of two targeting rules. The first instructs the policymaker how to set capital requirements in response to an opening up of the consumption and housing gap κ t = Φ c c t + Φ h ht, (18) where Φ c and Φ h are coefficients defined in the appendix, where we also demonstrate that both coefficients are positive for empirically plausible parameterizations of the model. Rule (18) is static and requires the policymaker to increase capital requirements whenever either a consumption gap or a housing gap (or both) opens up. Positive consumption or housing gaps signal excess demand by borrowers, and hence require a tightening of financial conditions. The policymaker achieves this goal by raising capital requirements, thus reducing credit and making it more expensive for borrowers. The second rule is dynamic and forward looking c t + Ω h ht = Ω c E t c t+1, (19) where Ω h and Ω c are coefficients defined in the appendix. Given that the static targeting rule sets capital requirement, we can think of rule (19) as implicitly determining the optimal LTV ratio. In particular, the combination of current consumption and housing gaps on the left-hand side must be proportional to the expected future consumption gap. Through the effects on the composite state variable (and in particular debt, but also current interest rates and capital requirements) optimal policy affects future losses. Under discretion, the policymaker cannot commit to any policy in the future. Therefore, current decisions need to take into account the effects on future outcomes. For example, if a negative shock hits the economy in the current period and opens up the consumption and housing gaps, the policymaker should relax capital requirements to minimize its macroeconomic impact. However, once the shock dies out, such an expansionary policy will contribute to overheat the economy. The policymaker should use the LTV ratio to ensure that current policy is not excessively costly in the future by allowing for excessive leverage. The example is purely illustrative and does not necessarily mean that the policymaker should always adjust the macro-prudential instruments in opposite directions. The targeting rules derived above illustrate that it is not generally possible for macro- 13

14 prudential policy to achieve full stabilization: that is, a stable equilibrium with c t = h t = κ t, t. We can demonstrate this using an informal proof by contradiction. If it was possible to deliver such an allocation using these targeting rules, then those rules also imply that κ t = (from (18)) and E t c t+1 = (from (19)). These conditions (together with the conjectured full stabilization allocations) can be substituted into the Euler equation and the real version of (17) to give: S t = βs 1 S t 1 + σνt c which implies an explosive trajectory for the composite state variable, since β s < 1. In the next section we shall see how the features of optimal macro-prudential policy under discretion in the efficient equilibrium extend to the case with nominal rigidities and interact with the optimal conduct of conventional monetary policy. 3.4 Monetary and Macro-Prudential Policies with Sticky Prices As in the previous section, we report here the optimal targeting rules for monetary and macroprudential policy under discretion, and refer to Appendix E.2 for the details of the derivation. The first result is that the introduction of sticky prices does not change the nature of the static tradeoffs for the macro-prudential authority. Equation (18) continues to describe the optimal setting of capital requirements. 11 A second static targeting rule characterizes optimal monetary policy x t + γλ π π t + Λ c c t =, (2) where Λ c is a coefficient defined in the appendix. Equation (2) resembles the standard New Keynesian monetary targeting rule under discretion, but also includes an adjustment for the consumption gap. Since the coefficient on the consumption gap Λ c is positive for empirically relevant calibrations, monetary policy will typically lean against the wind. Everything else equal, a shock that opens a positive consumption gap requires a negative combination of the output gap and inflation (appropriately weighted), contrary to the standard case (Clarida et al., 1999) in which the same combination should be set equal to zero. prices The last targeting rule extends the optimal setting of the LTV ratio to the case of sticky c t + Υ x x t + Υ π π t + Υ h ht = Υ c E t c t+1, (21) where Υ x, Υ π, Υ h, and Υ c are coefficients defined in the appendix. With sticky prices, the optimal setting of LTV ratios needs to take into account also current inflation and output gap. As in the efficient equilibrium, the current policy decisions affect future outcomes and losses through the composite state variable. A negative shock that hits the economy in the current period not only creates distributional effects, opening up a consumption and housing gap, but also negatively affects inflation and the output gap. The policymaker should respond to the shock by relaxing monetary and financial conditions, without losing sight of effects of the current 11 Formally, in the appendix we show that a subset of the first order conditions for the problem with sticky prices are identical to the static first order conditions of the case with flexible prices and real debt. Therefore, we can derive the same static macro-prudential rule. 14

15 policy response on future outcomes and losses. A simple extension of the argument in Section 3.3 shows that full stabilization is generally not possible, even in the absence of cost-push shocks. In sum, the rules (18), (2), and (21) reveal a rich interaction between monetary and macroprudential policy that operates via the output gap and inflation on the monetary side, and the consumption and housing gap on the macro-prudential side. The next section makes the targeting criteria above operational in the context of a house price boom scenario that mimics some aspects of the recent crisis. 4 Quantitative Experiments In this section, we use our model to study the interaction of monetary and macro-prudential policies in a stylized simulation of a house price boom. Our objective is to explore the interaction of monetary and macro-prudential policies under alternative policy configurations, when accounting for the presence of occasionally binding constraints. We consider several policy configurations, specifying which policy instruments may be used, which policymakers (monetary and/or macro-prudential) may use them and the objectives of the policymaker(s). Occasionally binding constraints are of interest because policy instruments (the short-term nominal interest rate and capital requirements) may be constrained by a lower bound and because the borrowing constraint may become slack. Both of these cases will change the ability of policy to stabilize the economy. To provide a somewhat more realistic dynamic structure of the model, we incorporate a slow-moving borrowing limit in the same way as Guerrieri and Iacoviello (217) and Justiniano et al. (215). This modification is intended to generate more persistent movements in debt and its marginal value (that is, the multiplier µ). In principle, it is possible to incorporate a wide range of additional frictions to enhance the dynamic properties of the model (as Guerrieri and Iacoviello, 217, do, for example). Here, we focus on the slow-moving borrowing limit largely because it does not affect the derivation of the welfare-based loss function while introducing some quantitative relevance. Specifically, we assume that borrowers face the following borrowing constraint: D b t(i) γ d D b t 1(i) + (1 γ d ) Θ t Q t H b t (i) (22) where γ d, 1) is a parameter controlling the extent to which the debt limit depends on the household s debt in the previous period. As argued by Guerrieri and Iacoviello (217), this formulation can be interpreted as capturing the idea that only a fraction of borrowers experience a change to their borrowing limit each period (which may be associated with moving or re-mortgaging). One implication of this formulation is that movements in debt adjust only gradually to changes in the value of the housing stock, which is consistent with the data in Figure 1. The modification to the specification of the borrowing limit affects the Euler equation and housing demand equation of borrowers as shown in Appendix F.1. When γ d = and Θ = 1 the model collapses to the version discussed in Section

16 Description Value Comments/issues/questions β s Saver discount factor.995 Guerrieri and Iacoviello (217) σ Inverse elasticity of substitution (consumption) 1 Guerrieri and Iacoviello (217) ϕ Inverse Frisch elasticity 1 Guerrieri and Iacoviello (217) γ d Debt limit inertia.7 Guerrieri and Iacoviello (217) Θ Debt limit (fraction of house value).9 Guerrieri and Iacoviello (217) γ Slope of Phillips curve.24 Eggertsson and Woodford (23) β b Borrower discount factor.99 See text. ξ Fraction of borrowers in economy.57 Cloyne et al. (216) η Debt: GDP ratio 1.8 BIS data (199 2) ψ Elasticity of funding cost to capital ratio.5 See text. σ h Inverse elasticity of substitution (housing) 25 See text. ρ h Housing demand shock persistence.85 See text. 4.1 Parameter Values Table 1: Parameter Values The parameter values used for the simulation exercises are shown in Table 1. Most of the parameter values are taken from the careful estimation of a similar (though richer) model on US data by Guerrieri and Iacoviello (217). We focus discussion on the remaining parameters. We set β b =.99, implying that borrowers are less patient than implied by the estimate of.992 by Guerrieri and Iacoviello (217). The relative discount factors of borrowers and lenders are crucial for the extent to which changes in house price expectations cause the borrowing constraint to become slack. Our lower value of β b increases the steady state value of the borrowing constraint multiplier so that larger shocks are required for the constraint to become slack. We set the slope of the Phillips curve in line with the assumption in Eggertsson and Woodford (23). 12 Two parameters that are important in determining the response to housing demand shocks are the persistence of the shock and the intertemporal substitution elasticity. We assume a moderate level of persistence, setting ρ h =.85, which is somewhat smaller than the modal estimate of.98 from Guerrieri and Iacoviello (217). We set σ h = 25 which implies that housing demand is rather insensitive to movements in the real house price. Guerrieri and Iacoviello (217) assume σ h = 1, but also incorporate habit formation in the sub-utility function for housing. The high degree of habit formation that they estimate implies that the short-run elasticity of housing demand to changes in the house price is much lower than unity. By setting σ h = 25, we aim to replicate this qualitative behavior without complicating the model and particularly the derivation of the welfare-based loss function. 13 The remaining parameters are set with reference to UK data. To set ξ we refer to the analysis in Cloyne et al. (216), who study the behavior of households by tenure type. Their data imply that UK household shares are roughly: 3% homeowners; 4% mortgagors and 3% renters. Since our model does not include renters, we set ξ = so that it represents 12 This requires a Calvo price adjustment parameter of.8875, a little lower than the estimate of Guerrieri and Iacoviello (217). 13 Of course, our approach also reduces the long-run elasticity of housing demand to house prices, so that it is less flexible than the introduction of habit formation. 16

17 the relative population shares of mortgagors and homeowners in the data. 14 We set η with reference to UK household debt to GDP ratios. According to BIS data, this ratio averaged around 6% between 199 and Around three quarters of household debt is mortgage debt, which suggests setting η = (since η is the ratio of debt to quarterly GDP). We assume the steady state capital ratio κ is 1%, close to the average reported in Meeks (217) for UK banks over the period Given that, the key determinant of the transmission of changes in κ t through to credit spreads is the parameter ψ Ψ κ. In its final report to the BIS, Macroeconomic Assessment Group (21) estimate that a 1 percentage point rise in capital requirements would have a peak effect on GDP of between -.5% and -.35%. A partial equilibrium thought experiment implies that this effect would be generated by an increase in (annualized) short-term nominal interest rates of between.2pp and 1.4pp when σ = 1, as in our calibration. However, in our model, credit spreads are only faced by about 6% of households, so the spread would need to increase by around.3 2.3pp on an annualized basis. We assume that the change in spreads is 2 percentage points, towards the top of this range. Taken together, these assumptions imply a value of Ψ solving (.2/4) = Ψ.1.1, or equivalently, a value of ψ given by (.2/4)/.1 = Simulation Methodology Our simulation is designed to generate a prolonged rise in the real price of housing followed by a sharp fall. The simulation is calibrated to deliver a similar increase in real house prices that was observed in pre-crisis UK data (Figure 1). The fall in prices is much more extreme than observed in the United Kingdom because our aim is to generate a sufficiently large downturn that monetary policy may be constrained by the zero bound purely as a consequence of the house price fall. In this respect, perhaps, the simulation more closely resembles the US experience. To generate a steady increase in house prices, we apply a sequence of unanticipated shocks to housing preferences u h t, t = 1,..., K. In each period, we solve the model (applying the occasionally binding constraints when necessary) conditional on agents information up to that date. For the boom phase, the sequence of shocks is increasing, so that u h t > u h t 1 for t = 1,..., K 1. The bust occurs in period K: a large negative housing preference shock is realised, which acts to reverse most of the previous increase in house prices. To match the slow pre-crisis increase in house prices we set assume that the boom lasts for 3 quarters (K = 31). We use a piecewise linear solution approach to account for the possibility that (a) the shortterm nominal interest rate is constrained by the zero lower bound and/or (b) borrowers debt limit (22) does not bind. This approach takes account of the possibility that the occasionally binding constraints may apply in future periods, but does not account for the risk that future shocks may cause the constraints to bind. This means that our solution approach does not 14 Guerrieri and Iacoviello (217) estimate that the fraction of labor income accruing to borrowers is around.5. Since labor income is allocated in proportion to population share in our model, this suggests a similar value for ξ. 15 This period precedes the run up in house prices before the financial crisis. We use this period for our calibration as we aim to mimic the pre-crisis house price rise in our simulation. 16 Meeks (217) estimates that a 5 basis point rise in capital requirements raises mortgages spreads by 2-25 annualized basis points at its peak. This implies a somewhat lower value of ψ =

18 Policy variant Policy assumptions Monetary Macro-prudential Flexible inflation targeting minimize L F IT Inactive (θ t = κ t = ) Leaning against the wind minimize L Inactive (θ t = κ t = ) Macro-prudential leadership minimize L F IT minimize L MaP Full co-ordination minimize L Table 2: Alternative policy assumptions used in simulations account for the skewness that may be generated in the expected distribution of future outcomes (e.g. of output and inflation) by the possibility of being constrained in future. Our solution methodology is therefore similar to the OccBin toolkit developed by Guerrieri and Iacoviello (215). Appendix F.4 contains a detailed description of our method. 4.3 Alternative Policy Configurations In our simulations, we study the macroeconomic effects of a housing boom under several alternative assumptions about the conduct of monetary and macro-prudential policies. These assumptions are motivated by the nature of central bank remits in the past and the way that the introduction of new macro-prudential policy instruments may affect those remits in the future. However, a common assumption that policy is set to minimize a loss function subject to constraints on their actions, as advocated in the context of monetary policy by Svensson (1999). Our alternative assumptions are based around the following decomposition of the welfarebased loss function given in equation (8): L E βs t ( x 2 t + λ π πt 2 ) + E t= } {{ } L F IT t= β t s (λ κ κ 2 t + λ c c 2 t + λ h h2 t ) } {{ } L MaP (23) The first component of the loss function, L F IT, is intended to capture the objectives encoded in the flexible inflation targeting mandates of many central banks in the pre-crisis period. The second component, L MaP, captures macro-prudential considerations. There are many potential ways to allocate objectives to different policymakers. For example, de Paoli and Paustian (217) examine a case in which the concern for output stabilization is divided between the monetary policymaker and macro-prudential policymaker. Our decomposition is based on the observation that if we take the limiting case in which the share of borrowers collapses to zero (ξ ) and (hence) the banking sector disappears (Ψ ), the model collapses to a standard New Keynesian model in which the only friction arises from price stickiness and the only policy instrument is the short-term nominal interest rate. So the existence of the financial frictions in our model generates both additional terms in the loss function and additional instruments with which to address them. On the basis of the decomposition of the welfare-based loss function in (23), Table 2 details the set of policy assumptions (or delegation schemes) that we consider in the next subsections. We explain the motivation and present the results for each variant in turn. We assume that the policymakers set policy in a time-consistent manner and are therefore 18